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Pseudo-Casimir forces in nematics with disorders in the bulk To cite this article: Fahimeh Karimi Pour Haddadan 2016 J. Phys.: Condens. Matter 28 405101
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- Fluctuation-induced forces in nematics with a foreign anisotropy in the bulk Fahimeh Karimi Pour Haddadan - Fluctuation-induced interactions in nematics with disordered anchoring energy Fahimeh Karimi Pour Haddadan, Ali Naji, Nafiseh Shirzadiani et al. - Pseudo-Casimir interactions across nematic films with disordered anchoring axis Fahimeh Karimi Pour Haddadan, Ali Naji, Azin Khame Seifi et al.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 28 (2016) 405101 (6pp)
doi:10.1088/0953-8984/28/40/405101
Pseudo-Casimir forces in nematics with disorders in the bulk Fahimeh Karimi Pour Haddadan Faculty of Physics, Kharazmi University, Tehran 15815-3587, Iran E-mail: [email protected] Received 12 April 2016, revised 26 June 2016 Accepted for publication 18 July 2016 Published 18 August 2016 Abstract
A nematic liquid-crystalline slab is considered in which some rod-like particles are randomly distributed. The particles are locally elongated either homeotropic or planar with respect to the confining substrates of the cell. We consider thermal fluctuations of a nematic director which is aligned perpendicular to the confining substrates due to strong homeotropic anchoring at the substrates. The resulting fluctuation-induced force across the cell is analyzed for an annealed disorder in the anchoring of the nematic director at the dispersed mesoscopic particles. Within the saddle-point approximation to free energy of the system, the effect of the disorder is renormalization of the strength of the mean anchoring which is assumed to be homeotropic. By increasing the variance of the disorder, the modes become less massive and deviations from the mean behavior become larger, so that the disorder-free universal long-range attraction, due to the soft modes, is approached. Keywords: liquid crystals, Casimir effect, disorder (Some figures may appear in colour only in the online journal)
1. Introduction
interaction. An annealed (thermalized) disorder does not create a new component and just enters into the force expression by renormalizing the mean value of the corresponding disordered parameter. But, on the other hand, if the fluctuations of the disorder occur in time scales much larger than the scale of thermal fluctuations, i.e. for a quenched (frozen) disorder, the disorder-induced force is completely decoupled from the thermal-induced component of the force. These are shown in the context of both electromagnetic [25–27] and liquid-crystalline Casimir forces [21, 22]. In addition to the bounding conditions, the range of correlations of the fluctuating field has also a decisive role in determination of the Casimir force. The effects of a constant external magnetic field [15, 28] and also an internal chiral field [29] which as well as the surface interactions influence the orientational order have been already studied and the behavior of the force ia analyzed. Here we would theor etically like to address how an inhomogeneity in the bulk of nematic can affect the force. We introduce a disorder in a nematic film by dispersions of rod-like mesoscopic particles throughout the nematic with a director aligned by
Confinements of liquid crystals [1] create surface energies (or boundary conditions) which affect not only the thermodynamics of the material but also may give rise to structural changes due to the elasticity of the liquid crystals. The later effect may, in turn, create a force between the confining objects [2]. For geometries where the orientational order remains on average uniform, thermal fluctuations can yet create a force, known as the pseudo-Casimir effect. This force [3–5], which generally arises whenever a correlated medium is confined [6], is considerably affected by the confinement conditions [7–18]. The effects of disorder (or inhomogeneity) on the bounding substrates (or interfaces) of a nematic medium have been considered in the studies of the thermodynamical and structural phase transitions [19, 20] as well as in the modifications of the pseudo-Casimir effect [21, 22]. Depending on the nature of the disorder, which can in general be annealed or quenched [23, 24] (or partially annealed), different behaviors have been suggested for this fluctuation-induced Casimir-like
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F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
strong homeotropic anchoring on the confining substrates of the film. In such a geometry, however, the fluctuations of a nematic director are locally either suppressed or enhanced due to the tendency of the local director to be in the direction of the rods which are supposed to impose local anchoring fields [30]. We model this interaction by an anchoring energy per unit volume W(r) which may vary over the space r. This model can, for instance, be manifested by a dispersion of carbon nanotubes (CNTs) in nematic liquid crystal [30–32] where depending on the model parameter of the composite, such as the ratio of nanotubes diameter to the nematic extrapolation length, the CNTs can align either parallel or perpend icular to the nematic director [31]. We consider a composite of parallel alignment on average with an experimentally unavoidable possibility for some local perpendicular alignment for CNTs. In addition to the disorder in the orientation, we assume a degree of disorder in anchoring strength as well by assuming that the number density of the rods may have a spatial variation. Our interest here is to examine how thermal fluctuations of a mean uniform director field, influenced by an anchoring field in the bulk, affect the free energy of the nematic system. One should note that the disorder in the bulk can of course be originated by other sources, such as by a nonuniform magnetic field over the nematic cell (induced for instance by a suspension of elongated magnetic grains throughout the nematic [1] or by applying some external magnetic fields of different strength and orientation). Then the present methodology can accordingly be implemented, through the replacement of W(r) by the corresponding magnetic inhomogeneity. In an annealed model for the disorder, our anchoring field is thermalized and thus the free energy of the system is obtained from the disorder averaged partition function which does not in general depend on any specific configuration of W(r). However, in order to evaluate the effect of a nonlinear coupling on the free energy of the nematic film we use the mean field approximation to the anchoring field and obtain the saddle-point free energy. In the case where the mean anchoring is homeotropic, the inter-substrate fluctuationinduced force is in the form of a force due to fluctuations of a finite correlation length (massive fluctuations). The disorder results in a renromalization of the mean anchoring coherence length = K /W0 in a characteristic way, where K is the effective elastic constant of nematic and W0 is the mean anchoring energy per unit volume. So that the range of the force, determined by the effective anchoring coherence length, can then be tuned by variance of the disorder in the anchoring strength and also, at intermediate separations where the effect of the disorder turns out to be pronounced, by inter-substrate separation of the film. In section 2, we introduce our model and formulate the effect of the disorder on the functional free energy of our system. Then in section 3, the functional-integrals are preformed to treat the disorder effect on the free energy of the system. We proceed with determining the fluctuationinduced force and analyzing our results in section 4. We conclude our discussion in section 5.
Figure 1. The nematic cell in the presence of a disorder in the bulk. The disorder is introduced by the rods which are randomly dispersed. Orientations of the rods are either perpendicular or planar with respect to the confining walls which impose strong homeotropic anchoring, so that the nematic’s molecules are, on average, aligned perpendicular to the walls. The mean alignment of the rods will be assumed to be homeotropic as well.
2. The model and formalism We consider a nematic liquid-crystalline film of a thickness d. In the bulk of nematic there are particular rod-like mesoscopic particles randomly dispersed. The rods are assumed to be locally either homeotropic (perpendicular) or planar with respect to the confining walls of the film. However, the mean orientation is assumed to be homoetropic. The strong homeotropic anchoring is imposed on the both confining substrates, located at z = 0 and z = d of a Cartesian coordinate system r = (x, z ), so that the mean director n 0 is aligned by the substrates, i.e. n 0 = zˆ . Namely, we consider a situation where the substrate-aligned configuration wins over the alignment dictated by our nonuniform (orientational and positional) distribution of the rods (figure 1), so the interaction of the nematic order and the rods is sufficiently weak, in order to avoid any structural transition in the nematic film. By treating each rod as an agent which tries to align locally the director along itself, we define an easy local direction, which is in the direction of the rod axis, for the director. Then the coupling of nematic director n(r) and the rods can be modeled, according to the free energy density of the quadratic form of the Rapini–Popoular model [33], as γ Faf = − ∑ drδ (r − Ri)[n(r) ⋅ e(r)]2 (1) 2 i
∫
where e(Ri) is the easy direction at the position Ri of the ith rod and γ is the interaction energy of the rod. Using a coarsegrained continuum density of the rods ρ(r) = ∑ i δ (r − Ri), the interaction reads 1 drW(r)[n(r) ⋅ e(r)]2 , Faf = − (2) 2 V
∫
where W(r) = γρ(r) is the anchoring energy per unit volume and the integral is taken over the volume of the film, V. This form is, indeed, a reinterpretation of the result in modeling the interaction term in nematic elastomers [34]. We assume and will use later that W is an annealed variable and consider a Gaussian distribution for it around a mean strength W0, with a 2
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
variance g which may in general have a temperature depend ence [22]. Based on Frank elastic theory [1], which gives the energy cost of the distortions of the director field,
Feff = ∑ ∑
K dr([∇ ⋅ n(r)]2 + [∇ × n(r)]2 ), Fn = (3) 2 V
∫
⎛ W⎞ ⎟ | ni (p, z ) |2 + ⎜p 2 + ⎝ K⎠
)
(7)
By performing the functional integrals over nx and ny, subject to the Dirichlet boundary conditions, the partition function of the system, ∫ dW P[W ]Z[W ], for W ⩾ 0 is given by [36]
∫
where K is the effective elastic constant in the one-constant approximation and nx, ny are components of n in the lateral space x = (x, y ). By assuming the easy direction e = zˆ , the interaction of n with the anchoring field in the bulk, equation (2), is given by
〈Z〉 =
1 drW(r)[n2x(r) + n2y(r)], Faf = (5) 2 V
∫
∫
⎛ 1 dW exp ⎜⎜− 2g ⎝
⎞
2⎟
∫ dr(W − W0) ⎟
⎡ sinh( p 2 + W/K d ) ⎥ , ⎢⎣ βKA p 2 + W/K ⎥⎦
∏⎢ p
where W (r) can take both positive and negative values corresponding to local homeotropic or planar anchoring axes, respectively. The partition function of the system is given by
⎤−1
⎠
(8)
where ⟨ ⟩ denotes the disorder average. The case W < 0, which exhibits a frustrating geometry [15] and results in a structural transition on approaching a critical separation, is outside the scope of the present paper. Now at this stage by using the saddle-point value of W we obtain the free energy, −kBT ln〈Z 〉, in the saddle-point approximation [35]. The saddle-point value has the largest contribution in the averaged partition function since it minimizes F,
∫
Z[W(r)] = Dnx Dn y exp[−β (Fn + Faf )], (6)
where β = 1/(kBT ), kB is the Boltzmann constant, and T is the temperature. The functional integrals over all possible configurations of the director fluctuations nx (r) and n y(r) at all points r in space V [35] are subject to the boundary conditions ni(x, 0) = ni(x, d ) = 0, i = x, y (strong homeotropic anchoring on the plates). Our partition function is a functional of the annealed field W(r) and thus it is expected that by functional integrating over all possible realizations of W(r), weighted by its Gaussian probability distribution func1
(
KA |∂zni(p, z )|2 2
3. The free energy
K dr([∇nx (r)]2 + [∇n y(r)]2 ), Fn = (4) 2 V
(
p
dz
where A = V/d is the area of the confining substrate and we have also used n j (p, z ) = nj (−p, z ).
the functional free energy of small thermal fluctuations of 1 1 the director n(r) = (nx (r), n y(r), 1 − 2 n2x(r) − 2 n2y(r)) up to second order is given by
∫
i, j
∫0
d
∂F = 0, (9) ∂W W = W
where F is given by βF (W, d ) =
)
tion, i.e. P[W ] = C exp − 2g ∫ dr(W(r) − W0)2 where C is the renormalization constant, this dependence is averaged out. However, this procedure results in a nonlinear term which cannot easily be tractable. To avoid this and in order to obtain a rough estimate of what may arise from such a geometry, we consider only the mean field contribution of our annealed anchoring field and obtain the saddle-point value of the field. On the other hand, in what follows W has no r dependence anymore and the effect of the disorder on the fluctuaioninduced interaction is evaluated within a saddle-point free energy. The mean anchoring is considered to be homeotropic in our model and thus W0 > 0 and the director fluctuations are on average suppressed. Translational invariance in the lateral space x lets us to use the Fourier expansion ni(x, z ) = ∑ p ni(p, z )e−ip ⋅ x, so that the effective functional free energy Feff = Fn + Faf is given by
V (W − W0)2 2g
⎛ sinh( p 2 + W/K d ) ⎞ ⎟. + ∑ ln⎜⎜ ⎟ 2 p β p + / K KA W ⎝ ⎠
(10)
The minimization of F with respect to W leads to ⎞ g 1 ⎛⎜ 1 Λ+ − d coth(ωd )⎟⎟ = 0, (11) ∑ ⎜ 2K W0V p ω ⎝ ω ⎠
where Λ = 1 − W /W0, ω =
p 2 + −2(1 − Λ) , and
= K /W0 , (12)
is the mean anchoring coherence length. Eventually the free energy of the system, within the saddle-point approximation, F (d ) = F (W , d ), (equation (10)), can thus be obtained as
3
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
βF (d ) =
⎛ sinh( p 2 + −2(1 − Λ) d ) ⎞ V 2 2 ⎟. (13) W 0Λ + ∑ ln⎜⎜ ⎟ 2 −2 2g β KA p + 1 − Λ ( ) p ⎝ ⎠
1.0 0.9
We remind that Λ should be kept below one (W ⩾ 0) in order to be in the regime of stabilized anchoring field.
0.8
3.1. Λ versus separation d
0.6
0.7
In order to obtain Λ as a function of the separation d we solve the saddle-point equation given by relation (11). We first use pmax pdp /(2π ), the continuum limit so that we replace ∑ p by A ∫ 0 where pmax = π /a is the ultra-violet cut-off and a is the molecular length. Then after performing the integration, Λ as a function of the rescaled separation d˜ = d / is determined by ⎡ ⎛ ⎢ sinh⎜ ⎝ χ Λ = ⎢ln d˜ ⎢ ⎢ ⎣
π 2 2 a2 π 2 2 a2
⎞ + (1 − Λ) d˜⎟ ⎠
+ (1 − Λ )
0.5 0.4
1.0
1.5
2.0
2.5
3.0
3.5
4.0
d W as a function of W0 g /(8π W 20 3) = 0.002 (triangles),
Figure 2. The numerical values of Λ = 1 −
⎤ sinh (1 − Λ) d˜ ⎥ ⎥, (14) − ln ⎥ (1 − Λ ) ⎥ ⎦
(
0.5
)
the rescaled separation d˜ = d / at 0.003 (squares) and /a = 100. The solid curves show the analytical estimates for the function Λ(d˜ ) given in equation (15).
∂F ∂F = ∂d ∂d
where χ = g /(8π W 20 3). For further analytical estimations, we let π /a 1 and first obtain the approximate value of Λ in the limiting values of d˜. In the limit of d , we obtain
+ Λ
∂F ∂Λ
d
∂Λ . ∂d
However, within the saddle-point approximation (equation (9)) the later term has no contribution to the force F and thus
χ sinh(πd /a ) Λ ln . (15) d πd /a
β F(d ) = −
By Taylor expansions of the functions around the asymptotically small rescaled separation d˜ → 0 in the above equation, the relation yields Λ χ 2p2max d˜ /6.
=−
∂βF ∂d ∂βFC ∂d
Λ
− ∑ ω,
(18)
p
Λ
where
At large separations, d , Λ behaves as
βFC = ∑ ln (1 − e−2ωd ). (19)
χ sinh(πd /a ) χπ /a, Λ ln (16) π /a d˜
p
With regard to the free energy of the reference bulk, assuming fluctuations around a uniform director along the mean orientation of the dispersed rods, a constant force equal to −kBT ∑ p ω is expected to arise. This bulk contribution can cancell the corresp onding contribution in equation (18). Thus the interaction force in the presence of the bulk disorder is given merely by making a d-derivative of FC, the modified pseudo-Casimir free energy
so that it saturates to a constant value. We note that, the relation (15) gives for Λ the same saturation value in the limit of πd /a 1, which is in fact the condition for the validity of the continuum elastic theory applied here. Thus it is expected that equation (15) gives a good estimate for the function Λ in the whole range of d˜. Equation (14) is numerically solved and the results for Λ as a function of d˜ for two different choices of the variance g are shown in figure 2. In this figure the function Λ given by equation (15) is also plotted which is clearly in agreement with the numerical data (symbols) for the chosen model parameters.
⎛ βFC = ∑ ln⎜⎜1 − e−2 ⎝ p
⎞ ⎟, eff = ⎠
2 p 2 + − eff d ⎟
1−Λ
,
at fixed Λ. The mean anchoring coherence length , as introduced by the mean value of the anchoring strength (equation (12)), is renormalized such that for g = 0 (or χ = 0), which gives Λ = 0 (equation (14)), the mean behavior is recovered. According to this, the pseudo-Casimir force FC reads
4. The fluctuation-induced force The variation of the free energy F(d) (equation (13)) with inter-surface separation d creates a force F across the film. To obtain this, we first decompose F as V 2 2 βF (d ) = W 0 Λ + ∑ [ln(1 − e−2ωd ) + ωd − ln(2βKAω )]. 2g p (17)
p 2 + −eff2 (20) β FC = 2 ∑ . −2 2 p 1 − e2 p + eff d
Thus the annealed anchoring field in the nematic bulk, as a whole, changes the nature of our fluctuating modes from the two soft modes (of an infinite correlation length) to two massive modes with a finite correlation length eff .
We note that in addition to explicit d-dependence, the free energy implicitly depends on d through Λ = Λ(d ), so that the d-derivative of F is, in general, given by 4
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
The effect of a finite nematic correlation length on the fluctuation-induced force is already studied; in the vicinity of the isotropic-nematic phase transition [11] and in nematics in the presence of a stabilized external magnetic field [28]. In the following we examine the effect of the disorder variance introduced by Λ.
0.00
4.1. Force FC versus separation d
0.06
By using the continuum limit and a change of variable q = pd, the expression of the force FC (equation (20)) converts to
0.08
0.02 0.04
2 ∞ q q 2 + (1 − Λ)d˜ 1 β FC (21) = dq. A πd 3 0 1 − e2 q 2 + (1 −Λ)d˜ 2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
d
∫
Figure 3. The rescaled pseudo-Casimir pressures Π(d˜ ) = β FC 3/A (equation (26)) as a function of the re-scaled separation d˜ = d / at g /(8π W 20 3) = 0.002 (triangles), 0.003 (squares) and /a = 100. The curves (thin lines) show the corresponding pressures using the analytical estimate (equation (15)) for the parameter Λ. The solid thick curve shows the results when the variance g is set to zero (Λ = 0) and the dashed curve shows the universal long-range attraction obtained in the absence of the rods (equation (22)).
This form is useful in deriving the force at the limiting values of d˜ as follows. In the limit of small separations, d˜ → 0, FC reduces to ∞ q 2 dq ζ (3) β FC 1 , = =− (22) πd 3 0 1 − e2q A 4πd 3
∫
−3 where ζ (3) = ∑∞ = 1.202 0569…. Thus for d , the m=1 m force is universal and long range. In this regime → ∞, the anchoring is weak (W0 → 0), and Λ → 0 (equation (15)). Thus the anchoring field in the bulk does not play any role. At large separations, d˜ → ∞, by approximating FC as
disorder turns out to be the largest, by increasing the disorder variance, the modes become softer and the long-range attraction is approached. Also in the same figure we have plotted the rescaled pressure Π as a function of d˜ (the curves crossing the data points), using the analytical relation (15) for Λ, which was turned out to be an acceptable function describing Λ in a wide extent of d˜.
∞ β FC 1−Λ 2 ˜2 (23) q e−2 q + (1 −Λ)d dq, − 2 0 A π d
∫
we obtain a short-range force given approximately by
5. Conclusion
β FC 1 − Λ −2 (1 −Λ) d˜ (24) e . − A 2π 2d
In the continuation of the works on the role of disorder on the fluctuation-induced forces in nematics [21, 22], we considered in this paper the influence of a thermalized (annealed) anchoring field in the bulk of a nematic cell on the thermal fluctuations of a uniform director field. In our model the anchoring energy per unit volume can take all values from −∞ to +∞ where the anchoring axis is locally either homeotropic or planar with respect to the mean director. A homeotropically aligned mean anchoring (with a mean strength W0 ) suppresses on average the fluctuations of a uniform homeotroipc director orientation and thus the fluctuation-induced force is short range and decays exponentially with the inter-plate separation d. The range of the force, eff , is reminiscent of Debye or Thomas–Fermi screening length κ−1 in screened Van der Waalds force, where on the basis of the Lifshitz theory, the zero-frequency (high temperature) contribution to the force per area between two flat substrates of static dielectric constant
We note that in this limit the effect of the disorder variance cannot be pronounced as well, since for → 0, Λ vanishes (equation (16)). We should also mention that with the use of −2mq 1/(1 − e2q) = − ∑∞ , the exact analytical form of the m=1 e force in terms of a series expansion is given by 2 β FC 1 ∞ 1 ⎡1 md ⎛ md ⎞ ⎤ − 2md ⎢ + ⎥ e eff , =− + ⎜ ⎟ (25) ∑ A 2πd 3 m = 1 m3 ⎢⎣ 2 eff ⎝ eff ⎠ ⎥⎦
which reproduces the same limiting behaviors as above [15, 28]. The rescaled pseudo-Casimir pressure Π = β FC 3/A, 1 ∞ q q 2 + (1 − Λ) (26) Π= dq, π 0 1 − e2 q 2 + (1 −Λ) d˜ as a function of the rescaled separation d˜ is numerically evaluated, using the numerically evaluated values of Λ obtained in the previous section 3.1, for the whole range of d˜. The results (symbols) at two different variances are compared with the field-free case, i.e. the long-range attraction (dashed curve) and the attraction due to modes with a finite correlation length (equation (26) for Λ = 0 (g = 0)) (solid curve) in figure 3. In the regime of intermediate separations, where the effect of the
∫
ε3 ε2 − ε3 ε1 and ε 2 approximately reads − 43 6kπBdT3 εε1 − (2κd )e−2κd, 1 + ε3 ε2 + ε3 where ε 3 is the dielectric constant of the intervening medium [37]. To deal with disorder in our nonlinear interacting model (equation (5)), we take recourse to the saddle-point approximation which gives the largest contribution of the disorder in the free energy of the system. According to this, the mean anchoring coherence length = K /W0 (with K 5
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
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the effective elastic constant) is renormalized and an effective nematic correlation length eff = / 1 − Λ is introduced. Λ scales with variance g (with dimension of (energy)2/(length)3) of the anchoring strength and increases with the inter-substrate separation d up to a constant value given by g 2/(8aK2 ) with a the short-distance cut off. Thus the rang of the force can, in principle, be varied (in a limited range) by varying the variance and the separation. For a finite , the force lies in a region which is bounded from below by the long-range force (∝ −1/d 3), which is reached for Λ → 1, and from above by the short-ranged force [∝−e−2d / /( 2d )], achieved for Λ → 0 (figure 3). We note that by decreasing , the fluctuationinduced force decreases and eventually when the anchoring coherence length becomes zero (or equivalently for a strong pinning, W0 → ∞), the force vanishes altogether. The model system for the disorder here is the same as the model considered in [22], with a distinction that here our dis order is extended into the bulk of the nematic, whereas in [22], the disorder is confined to one of the substrates of the cell, which is a 2 dimensional (2D) flat surface. Regardless of the differences in the boundary conditions, and thus the fluctuationinduced forces, the effect of the disorder is summarized in an effective parameter as Weff = W0(1 − Λ). It is interesting that the behavior of Λ versus d, in the both case, in a wide extent, is rather similar. However, for asymptotically small separations d, although Λ similar to the 2D case increases linearly with d, its magnitude is 1/3 of the corresponding value in the 2D case. Upon increasing the separation, the variation of Λ is smoother here than in 2D. Then for d , Λ saturates in the both case. The interest in dispersed particles in liquid crystals [38, 39], such as carbon nanotubes [30–32, 40] for example, may create some further motivations for the studies of the liquid-crystalline Casimir forces in the presence of non-isotropic particles, which might find applications in nanotechnologies as well. References [1] de Gennes P G and Prost J 1995 The Physics of Liquid Crystals (Oxford: Oxford Science Publications) [2] Eskandari Z, Silvestre N M, Tasinkevych M and Telo da Gama M M 2012 Soft Matter 8 10100 [3] Parsegian V A 2005 Van der Waals Forces (Cambridge: Cambridge University Press) [4] Bordag M, Klim-chitskaya G L, Mohideen U and Mostepanenko V M 2009 Advances in the Casimir Effect (New York: Oxford University Press) [5] Mostepanenko V M and Trunov N N 1997 The Casimir Effect and Its Applications (Oxford: Oxford University Press) [6] Kardar M and Golestanian R 1999 Rev. Mod. Phys. 71 1233 [7] Ajdari A, Peliti L and Prost J 1991 Phys. Rev. Lett. 66 1481 [8] Ajdari A, Duplantier B, Hone D, Peliti L and Prost J 1992 J. Phys. II 2 487
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Pseudo-Casimir forces in nematics with disorders in the bulk To cite this article: Fahimeh Karimi Pour Haddadan 2016 J. Phys.: Condens. Matter 28 405101
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 28 (2016) 405101 (6pp)
doi:10.1088/0953-8984/28/40/405101
Pseudo-Casimir forces in nematics with disorders in the bulk Fahimeh Karimi Pour Haddadan Faculty of Physics, Kharazmi University, Tehran 15815-3587, Iran E-mail: [email protected] Received 12 April 2016, revised 26 June 2016 Accepted for publication 18 July 2016 Published 18 August 2016 Abstract
A nematic liquid-crystalline slab is considered in which some rod-like particles are randomly distributed. The particles are locally elongated either homeotropic or planar with respect to the confining substrates of the cell. We consider thermal fluctuations of a nematic director which is aligned perpendicular to the confining substrates due to strong homeotropic anchoring at the substrates. The resulting fluctuation-induced force across the cell is analyzed for an annealed disorder in the anchoring of the nematic director at the dispersed mesoscopic particles. Within the saddle-point approximation to free energy of the system, the effect of the disorder is renormalization of the strength of the mean anchoring which is assumed to be homeotropic. By increasing the variance of the disorder, the modes become less massive and deviations from the mean behavior become larger, so that the disorder-free universal long-range attraction, due to the soft modes, is approached. Keywords: liquid crystals, Casimir effect, disorder (Some figures may appear in colour only in the online journal)
1. Introduction
interaction. An annealed (thermalized) disorder does not create a new component and just enters into the force expression by renormalizing the mean value of the corresponding disordered parameter. But, on the other hand, if the fluctuations of the disorder occur in time scales much larger than the scale of thermal fluctuations, i.e. for a quenched (frozen) disorder, the disorder-induced force is completely decoupled from the thermal-induced component of the force. These are shown in the context of both electromagnetic [25–27] and liquid-crystalline Casimir forces [21, 22]. In addition to the bounding conditions, the range of correlations of the fluctuating field has also a decisive role in determination of the Casimir force. The effects of a constant external magnetic field [15, 28] and also an internal chiral field [29] which as well as the surface interactions influence the orientational order have been already studied and the behavior of the force ia analyzed. Here we would theor etically like to address how an inhomogeneity in the bulk of nematic can affect the force. We introduce a disorder in a nematic film by dispersions of rod-like mesoscopic particles throughout the nematic with a director aligned by
Confinements of liquid crystals [1] create surface energies (or boundary conditions) which affect not only the thermodynamics of the material but also may give rise to structural changes due to the elasticity of the liquid crystals. The later effect may, in turn, create a force between the confining objects [2]. For geometries where the orientational order remains on average uniform, thermal fluctuations can yet create a force, known as the pseudo-Casimir effect. This force [3–5], which generally arises whenever a correlated medium is confined [6], is considerably affected by the confinement conditions [7–18]. The effects of disorder (or inhomogeneity) on the bounding substrates (or interfaces) of a nematic medium have been considered in the studies of the thermodynamical and structural phase transitions [19, 20] as well as in the modifications of the pseudo-Casimir effect [21, 22]. Depending on the nature of the disorder, which can in general be annealed or quenched [23, 24] (or partially annealed), different behaviors have been suggested for this fluctuation-induced Casimir-like
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F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
strong homeotropic anchoring on the confining substrates of the film. In such a geometry, however, the fluctuations of a nematic director are locally either suppressed or enhanced due to the tendency of the local director to be in the direction of the rods which are supposed to impose local anchoring fields [30]. We model this interaction by an anchoring energy per unit volume W(r) which may vary over the space r. This model can, for instance, be manifested by a dispersion of carbon nanotubes (CNTs) in nematic liquid crystal [30–32] where depending on the model parameter of the composite, such as the ratio of nanotubes diameter to the nematic extrapolation length, the CNTs can align either parallel or perpend icular to the nematic director [31]. We consider a composite of parallel alignment on average with an experimentally unavoidable possibility for some local perpendicular alignment for CNTs. In addition to the disorder in the orientation, we assume a degree of disorder in anchoring strength as well by assuming that the number density of the rods may have a spatial variation. Our interest here is to examine how thermal fluctuations of a mean uniform director field, influenced by an anchoring field in the bulk, affect the free energy of the nematic system. One should note that the disorder in the bulk can of course be originated by other sources, such as by a nonuniform magnetic field over the nematic cell (induced for instance by a suspension of elongated magnetic grains throughout the nematic [1] or by applying some external magnetic fields of different strength and orientation). Then the present methodology can accordingly be implemented, through the replacement of W(r) by the corresponding magnetic inhomogeneity. In an annealed model for the disorder, our anchoring field is thermalized and thus the free energy of the system is obtained from the disorder averaged partition function which does not in general depend on any specific configuration of W(r). However, in order to evaluate the effect of a nonlinear coupling on the free energy of the nematic film we use the mean field approximation to the anchoring field and obtain the saddle-point free energy. In the case where the mean anchoring is homeotropic, the inter-substrate fluctuationinduced force is in the form of a force due to fluctuations of a finite correlation length (massive fluctuations). The disorder results in a renromalization of the mean anchoring coherence length = K /W0 in a characteristic way, where K is the effective elastic constant of nematic and W0 is the mean anchoring energy per unit volume. So that the range of the force, determined by the effective anchoring coherence length, can then be tuned by variance of the disorder in the anchoring strength and also, at intermediate separations where the effect of the disorder turns out to be pronounced, by inter-substrate separation of the film. In section 2, we introduce our model and formulate the effect of the disorder on the functional free energy of our system. Then in section 3, the functional-integrals are preformed to treat the disorder effect on the free energy of the system. We proceed with determining the fluctuationinduced force and analyzing our results in section 4. We conclude our discussion in section 5.
Figure 1. The nematic cell in the presence of a disorder in the bulk. The disorder is introduced by the rods which are randomly dispersed. Orientations of the rods are either perpendicular or planar with respect to the confining walls which impose strong homeotropic anchoring, so that the nematic’s molecules are, on average, aligned perpendicular to the walls. The mean alignment of the rods will be assumed to be homeotropic as well.
2. The model and formalism We consider a nematic liquid-crystalline film of a thickness d. In the bulk of nematic there are particular rod-like mesoscopic particles randomly dispersed. The rods are assumed to be locally either homeotropic (perpendicular) or planar with respect to the confining walls of the film. However, the mean orientation is assumed to be homoetropic. The strong homeotropic anchoring is imposed on the both confining substrates, located at z = 0 and z = d of a Cartesian coordinate system r = (x, z ), so that the mean director n 0 is aligned by the substrates, i.e. n 0 = zˆ . Namely, we consider a situation where the substrate-aligned configuration wins over the alignment dictated by our nonuniform (orientational and positional) distribution of the rods (figure 1), so the interaction of the nematic order and the rods is sufficiently weak, in order to avoid any structural transition in the nematic film. By treating each rod as an agent which tries to align locally the director along itself, we define an easy local direction, which is in the direction of the rod axis, for the director. Then the coupling of nematic director n(r) and the rods can be modeled, according to the free energy density of the quadratic form of the Rapini–Popoular model [33], as γ Faf = − ∑ drδ (r − Ri)[n(r) ⋅ e(r)]2 (1) 2 i
∫
where e(Ri) is the easy direction at the position Ri of the ith rod and γ is the interaction energy of the rod. Using a coarsegrained continuum density of the rods ρ(r) = ∑ i δ (r − Ri), the interaction reads 1 drW(r)[n(r) ⋅ e(r)]2 , Faf = − (2) 2 V
∫
where W(r) = γρ(r) is the anchoring energy per unit volume and the integral is taken over the volume of the film, V. This form is, indeed, a reinterpretation of the result in modeling the interaction term in nematic elastomers [34]. We assume and will use later that W is an annealed variable and consider a Gaussian distribution for it around a mean strength W0, with a 2
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
variance g which may in general have a temperature depend ence [22]. Based on Frank elastic theory [1], which gives the energy cost of the distortions of the director field,
Feff = ∑ ∑
K dr([∇ ⋅ n(r)]2 + [∇ × n(r)]2 ), Fn = (3) 2 V
∫
⎛ W⎞ ⎟ | ni (p, z ) |2 + ⎜p 2 + ⎝ K⎠
)
(7)
By performing the functional integrals over nx and ny, subject to the Dirichlet boundary conditions, the partition function of the system, ∫ dW P[W ]Z[W ], for W ⩾ 0 is given by [36]
∫
where K is the effective elastic constant in the one-constant approximation and nx, ny are components of n in the lateral space x = (x, y ). By assuming the easy direction e = zˆ , the interaction of n with the anchoring field in the bulk, equation (2), is given by
〈Z〉 =
1 drW(r)[n2x(r) + n2y(r)], Faf = (5) 2 V
∫
∫
⎛ 1 dW exp ⎜⎜− 2g ⎝
⎞
2⎟
∫ dr(W − W0) ⎟
⎡ sinh( p 2 + W/K d ) ⎥ , ⎢⎣ βKA p 2 + W/K ⎥⎦
∏⎢ p
where W (r) can take both positive and negative values corresponding to local homeotropic or planar anchoring axes, respectively. The partition function of the system is given by
⎤−1
⎠
(8)
where ⟨ ⟩ denotes the disorder average. The case W < 0, which exhibits a frustrating geometry [15] and results in a structural transition on approaching a critical separation, is outside the scope of the present paper. Now at this stage by using the saddle-point value of W we obtain the free energy, −kBT ln〈Z 〉, in the saddle-point approximation [35]. The saddle-point value has the largest contribution in the averaged partition function since it minimizes F,
∫
Z[W(r)] = Dnx Dn y exp[−β (Fn + Faf )], (6)
where β = 1/(kBT ), kB is the Boltzmann constant, and T is the temperature. The functional integrals over all possible configurations of the director fluctuations nx (r) and n y(r) at all points r in space V [35] are subject to the boundary conditions ni(x, 0) = ni(x, d ) = 0, i = x, y (strong homeotropic anchoring on the plates). Our partition function is a functional of the annealed field W(r) and thus it is expected that by functional integrating over all possible realizations of W(r), weighted by its Gaussian probability distribution func1
(
KA |∂zni(p, z )|2 2
3. The free energy
K dr([∇nx (r)]2 + [∇n y(r)]2 ), Fn = (4) 2 V
(
p
dz
where A = V/d is the area of the confining substrate and we have also used n j (p, z ) = nj (−p, z ).
the functional free energy of small thermal fluctuations of 1 1 the director n(r) = (nx (r), n y(r), 1 − 2 n2x(r) − 2 n2y(r)) up to second order is given by
∫
i, j
∫0
d
∂F = 0, (9) ∂W W = W
where F is given by βF (W, d ) =
)
tion, i.e. P[W ] = C exp − 2g ∫ dr(W(r) − W0)2 where C is the renormalization constant, this dependence is averaged out. However, this procedure results in a nonlinear term which cannot easily be tractable. To avoid this and in order to obtain a rough estimate of what may arise from such a geometry, we consider only the mean field contribution of our annealed anchoring field and obtain the saddle-point value of the field. On the other hand, in what follows W has no r dependence anymore and the effect of the disorder on the fluctuaioninduced interaction is evaluated within a saddle-point free energy. The mean anchoring is considered to be homeotropic in our model and thus W0 > 0 and the director fluctuations are on average suppressed. Translational invariance in the lateral space x lets us to use the Fourier expansion ni(x, z ) = ∑ p ni(p, z )e−ip ⋅ x, so that the effective functional free energy Feff = Fn + Faf is given by
V (W − W0)2 2g
⎛ sinh( p 2 + W/K d ) ⎞ ⎟. + ∑ ln⎜⎜ ⎟ 2 p β p + / K KA W ⎝ ⎠
(10)
The minimization of F with respect to W leads to ⎞ g 1 ⎛⎜ 1 Λ+ − d coth(ωd )⎟⎟ = 0, (11) ∑ ⎜ 2K W0V p ω ⎝ ω ⎠
where Λ = 1 − W /W0, ω =
p 2 + −2(1 − Λ) , and
= K /W0 , (12)
is the mean anchoring coherence length. Eventually the free energy of the system, within the saddle-point approximation, F (d ) = F (W , d ), (equation (10)), can thus be obtained as
3
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
βF (d ) =
⎛ sinh( p 2 + −2(1 − Λ) d ) ⎞ V 2 2 ⎟. (13) W 0Λ + ∑ ln⎜⎜ ⎟ 2 −2 2g β KA p + 1 − Λ ( ) p ⎝ ⎠
1.0 0.9
We remind that Λ should be kept below one (W ⩾ 0) in order to be in the regime of stabilized anchoring field.
0.8
3.1. Λ versus separation d
0.6
0.7
In order to obtain Λ as a function of the separation d we solve the saddle-point equation given by relation (11). We first use pmax pdp /(2π ), the continuum limit so that we replace ∑ p by A ∫ 0 where pmax = π /a is the ultra-violet cut-off and a is the molecular length. Then after performing the integration, Λ as a function of the rescaled separation d˜ = d / is determined by ⎡ ⎛ ⎢ sinh⎜ ⎝ χ Λ = ⎢ln d˜ ⎢ ⎢ ⎣
π 2 2 a2 π 2 2 a2
⎞ + (1 − Λ) d˜⎟ ⎠
+ (1 − Λ )
0.5 0.4
1.0
1.5
2.0
2.5
3.0
3.5
4.0
d W as a function of W0 g /(8π W 20 3) = 0.002 (triangles),
Figure 2. The numerical values of Λ = 1 −
⎤ sinh (1 − Λ) d˜ ⎥ ⎥, (14) − ln ⎥ (1 − Λ ) ⎥ ⎦
(
0.5
)
the rescaled separation d˜ = d / at 0.003 (squares) and /a = 100. The solid curves show the analytical estimates for the function Λ(d˜ ) given in equation (15).
∂F ∂F = ∂d ∂d
where χ = g /(8π W 20 3). For further analytical estimations, we let π /a 1 and first obtain the approximate value of Λ in the limiting values of d˜. In the limit of d , we obtain
+ Λ
∂F ∂Λ
d
∂Λ . ∂d
However, within the saddle-point approximation (equation (9)) the later term has no contribution to the force F and thus
χ sinh(πd /a ) Λ ln . (15) d πd /a
β F(d ) = −
By Taylor expansions of the functions around the asymptotically small rescaled separation d˜ → 0 in the above equation, the relation yields Λ χ 2p2max d˜ /6.
=−
∂βF ∂d ∂βFC ∂d
Λ
− ∑ ω,
(18)
p
Λ
where
At large separations, d , Λ behaves as
βFC = ∑ ln (1 − e−2ωd ). (19)
χ sinh(πd /a ) χπ /a, Λ ln (16) π /a d˜
p
With regard to the free energy of the reference bulk, assuming fluctuations around a uniform director along the mean orientation of the dispersed rods, a constant force equal to −kBT ∑ p ω is expected to arise. This bulk contribution can cancell the corresp onding contribution in equation (18). Thus the interaction force in the presence of the bulk disorder is given merely by making a d-derivative of FC, the modified pseudo-Casimir free energy
so that it saturates to a constant value. We note that, the relation (15) gives for Λ the same saturation value in the limit of πd /a 1, which is in fact the condition for the validity of the continuum elastic theory applied here. Thus it is expected that equation (15) gives a good estimate for the function Λ in the whole range of d˜. Equation (14) is numerically solved and the results for Λ as a function of d˜ for two different choices of the variance g are shown in figure 2. In this figure the function Λ given by equation (15) is also plotted which is clearly in agreement with the numerical data (symbols) for the chosen model parameters.
⎛ βFC = ∑ ln⎜⎜1 − e−2 ⎝ p
⎞ ⎟, eff = ⎠
2 p 2 + − eff d ⎟
1−Λ
,
at fixed Λ. The mean anchoring coherence length , as introduced by the mean value of the anchoring strength (equation (12)), is renormalized such that for g = 0 (or χ = 0), which gives Λ = 0 (equation (14)), the mean behavior is recovered. According to this, the pseudo-Casimir force FC reads
4. The fluctuation-induced force The variation of the free energy F(d) (equation (13)) with inter-surface separation d creates a force F across the film. To obtain this, we first decompose F as V 2 2 βF (d ) = W 0 Λ + ∑ [ln(1 − e−2ωd ) + ωd − ln(2βKAω )]. 2g p (17)
p 2 + −eff2 (20) β FC = 2 ∑ . −2 2 p 1 − e2 p + eff d
Thus the annealed anchoring field in the nematic bulk, as a whole, changes the nature of our fluctuating modes from the two soft modes (of an infinite correlation length) to two massive modes with a finite correlation length eff .
We note that in addition to explicit d-dependence, the free energy implicitly depends on d through Λ = Λ(d ), so that the d-derivative of F is, in general, given by 4
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
The effect of a finite nematic correlation length on the fluctuation-induced force is already studied; in the vicinity of the isotropic-nematic phase transition [11] and in nematics in the presence of a stabilized external magnetic field [28]. In the following we examine the effect of the disorder variance introduced by Λ.
0.00
4.1. Force FC versus separation d
0.06
By using the continuum limit and a change of variable q = pd, the expression of the force FC (equation (20)) converts to
0.08
0.02 0.04
2 ∞ q q 2 + (1 − Λ)d˜ 1 β FC (21) = dq. A πd 3 0 1 − e2 q 2 + (1 −Λ)d˜ 2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
d
∫
Figure 3. The rescaled pseudo-Casimir pressures Π(d˜ ) = β FC 3/A (equation (26)) as a function of the re-scaled separation d˜ = d / at g /(8π W 20 3) = 0.002 (triangles), 0.003 (squares) and /a = 100. The curves (thin lines) show the corresponding pressures using the analytical estimate (equation (15)) for the parameter Λ. The solid thick curve shows the results when the variance g is set to zero (Λ = 0) and the dashed curve shows the universal long-range attraction obtained in the absence of the rods (equation (22)).
This form is useful in deriving the force at the limiting values of d˜ as follows. In the limit of small separations, d˜ → 0, FC reduces to ∞ q 2 dq ζ (3) β FC 1 , = =− (22) πd 3 0 1 − e2q A 4πd 3
∫
−3 where ζ (3) = ∑∞ = 1.202 0569…. Thus for d , the m=1 m force is universal and long range. In this regime → ∞, the anchoring is weak (W0 → 0), and Λ → 0 (equation (15)). Thus the anchoring field in the bulk does not play any role. At large separations, d˜ → ∞, by approximating FC as
disorder turns out to be the largest, by increasing the disorder variance, the modes become softer and the long-range attraction is approached. Also in the same figure we have plotted the rescaled pressure Π as a function of d˜ (the curves crossing the data points), using the analytical relation (15) for Λ, which was turned out to be an acceptable function describing Λ in a wide extent of d˜.
∞ β FC 1−Λ 2 ˜2 (23) q e−2 q + (1 −Λ)d dq, − 2 0 A π d
∫
we obtain a short-range force given approximately by
5. Conclusion
β FC 1 − Λ −2 (1 −Λ) d˜ (24) e . − A 2π 2d
In the continuation of the works on the role of disorder on the fluctuation-induced forces in nematics [21, 22], we considered in this paper the influence of a thermalized (annealed) anchoring field in the bulk of a nematic cell on the thermal fluctuations of a uniform director field. In our model the anchoring energy per unit volume can take all values from −∞ to +∞ where the anchoring axis is locally either homeotropic or planar with respect to the mean director. A homeotropically aligned mean anchoring (with a mean strength W0 ) suppresses on average the fluctuations of a uniform homeotroipc director orientation and thus the fluctuation-induced force is short range and decays exponentially with the inter-plate separation d. The range of the force, eff , is reminiscent of Debye or Thomas–Fermi screening length κ−1 in screened Van der Waalds force, where on the basis of the Lifshitz theory, the zero-frequency (high temperature) contribution to the force per area between two flat substrates of static dielectric constant
We note that in this limit the effect of the disorder variance cannot be pronounced as well, since for → 0, Λ vanishes (equation (16)). We should also mention that with the use of −2mq 1/(1 − e2q) = − ∑∞ , the exact analytical form of the m=1 e force in terms of a series expansion is given by 2 β FC 1 ∞ 1 ⎡1 md ⎛ md ⎞ ⎤ − 2md ⎢ + ⎥ e eff , =− + ⎜ ⎟ (25) ∑ A 2πd 3 m = 1 m3 ⎢⎣ 2 eff ⎝ eff ⎠ ⎥⎦
which reproduces the same limiting behaviors as above [15, 28]. The rescaled pseudo-Casimir pressure Π = β FC 3/A, 1 ∞ q q 2 + (1 − Λ) (26) Π= dq, π 0 1 − e2 q 2 + (1 −Λ) d˜ as a function of the rescaled separation d˜ is numerically evaluated, using the numerically evaluated values of Λ obtained in the previous section 3.1, for the whole range of d˜. The results (symbols) at two different variances are compared with the field-free case, i.e. the long-range attraction (dashed curve) and the attraction due to modes with a finite correlation length (equation (26) for Λ = 0 (g = 0)) (solid curve) in figure 3. In the regime of intermediate separations, where the effect of the
∫
ε3 ε2 − ε3 ε1 and ε 2 approximately reads − 43 6kπBdT3 εε1 − (2κd )e−2κd, 1 + ε3 ε2 + ε3 where ε 3 is the dielectric constant of the intervening medium [37]. To deal with disorder in our nonlinear interacting model (equation (5)), we take recourse to the saddle-point approximation which gives the largest contribution of the disorder in the free energy of the system. According to this, the mean anchoring coherence length = K /W0 (with K 5
F Karimi Pour Haddadan
J. Phys.: Condens. Matter 28 (2016) 405101
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the effective elastic constant) is renormalized and an effective nematic correlation length eff = / 1 − Λ is introduced. Λ scales with variance g (with dimension of (energy)2/(length)3) of the anchoring strength and increases with the inter-substrate separation d up to a constant value given by g 2/(8aK2 ) with a the short-distance cut off. Thus the rang of the force can, in principle, be varied (in a limited range) by varying the variance and the separation. For a finite , the force lies in a region which is bounded from below by the long-range force (∝ −1/d 3), which is reached for Λ → 1, and from above by the short-ranged force [∝−e−2d / /( 2d )], achieved for Λ → 0 (figure 3). We note that by decreasing , the fluctuationinduced force decreases and eventually when the anchoring coherence length becomes zero (or equivalently for a strong pinning, W0 → ∞), the force vanishes altogether. The model system for the disorder here is the same as the model considered in [22], with a distinction that here our dis order is extended into the bulk of the nematic, whereas in [22], the disorder is confined to one of the substrates of the cell, which is a 2 dimensional (2D) flat surface. Regardless of the differences in the boundary conditions, and thus the fluctuationinduced forces, the effect of the disorder is summarized in an effective parameter as Weff = W0(1 − Λ). It is interesting that the behavior of Λ versus d, in the both case, in a wide extent, is rather similar. However, for asymptotically small separations d, although Λ similar to the 2D case increases linearly with d, its magnitude is 1/3 of the corresponding value in the 2D case. Upon increasing the separation, the variation of Λ is smoother here than in 2D. Then for d , Λ saturates in the both case. The interest in dispersed particles in liquid crystals [38, 39], such as carbon nanotubes [30–32, 40] for example, may create some further motivations for the studies of the liquid-crystalline Casimir forces in the presence of non-isotropic particles, which might find applications in nanotechnologies as well. References [1] de Gennes P G and Prost J 1995 The Physics of Liquid Crystals (Oxford: Oxford Science Publications) [2] Eskandari Z, Silvestre N M, Tasinkevych M and Telo da Gama M M 2012 Soft Matter 8 10100 [3] Parsegian V A 2005 Van der Waals Forces (Cambridge: Cambridge University Press) [4] Bordag M, Klim-chitskaya G L, Mohideen U and Mostepanenko V M 2009 Advances in the Casimir Effect (New York: Oxford University Press) [5] Mostepanenko V M and Trunov N N 1997 The Casimir Effect and Its Applications (Oxford: Oxford University Press) [6] Kardar M and Golestanian R 1999 Rev. Mod. Phys. 71 1233 [7] Ajdari A, Peliti L and Prost J 1991 Phys. Rev. Lett. 66 1481 [8] Ajdari A, Duplantier B, Hone D, Peliti L and Prost J 1992 J. Phys. II 2 487
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